If ` vec x , vec y` are two non-zero and non-collinear vectors satisfying `[(a-2)alpha^2+(b-3)alpha+c] vec x+[(a-2)beta^2+(b-3)beta+c] vec y+[(a-2)gamma^2+(b-3)gamma+c]( vec xxx vec y)=0,w h e r ealpha,beta,gamma` are three distinct real numbers, then find the value of `(a^2+b^2+c^2-4)dot`

Explanation

To solve the problem step-by-step, we start with the given equation:

\[ [(a-2)\alpha^2 + (b-3)\alpha + c] \vec{x} + [(a-2)\beta^2 + (b-3)\beta + c] \vec{y} + [(a-2)\gamma^2 + (b-3)\gamma + c](\vec{x} \times \vec{y}) = 0 \]

### Step 1: Understanding the Equation
The equation states that a linear combination of the vectors \(\vec{x}\), \(\vec{y}\), and \(\vec{x} \times \vec{y}\) equals zero. Since \(\vec{x}\) and \(\vec{y}\) are non-zero and non-collinear, the only way for this equation to hold true is if the coefficients of each vector are equal to zero.

### Step 2: Setting Up the Coefficients
From the equation, we can derive three separate equations by setting the coefficients of \(\vec{x}\), \(\vec{y}\), and \(\vec{x} \times \vec{y}\) to zero:

1. \(a - 2\alpha^2 + b - 3\alpha + c = 0\) (for \(\vec{x}\))
2. \(a - 2\beta^2 + b - 3\beta + c = 0\) (for \(\vec{y}\))
3. \(a - 2\gamma^2 + b - 3\gamma + c = 0\) (for \(\vec{x} \times \vec{y}\))

### Step 3: Analyzing the Coefficients
We observe that all three equations have the same structure. This suggests that the coefficients of \(\alpha^2\), \(\alpha\), and the constant term must be equal across all three equations.

From the three equations, we can deduce:

- Coefficient of \(\alpha^2\): \(a - 2\)
- Coefficient of \(\alpha\): \(b - 3\)
- Constant term: \(c\)

### Step 4: Setting Coefficients to Zero
For the equations to hold, we set:

1. \(a - 2 = 0\) ⇒ \(a = 2\)
2. \(b - 3 = 0\) ⇒ \(b = 3\)
3. \(c = 0\)

### Step 5: Finding the Required Expression
Now we need to find the value of \(a^2 + b^2 + c^2 - 4\):

\[ a^2 + b^2 + c^2 - 4 = 2^2 + 3^2 + 0^2 - 4 \]

Calculating each term:

- \(2^2 = 4\)
- \(3^2 = 9\)
- \(0^2 = 0\)

So, we have:

\[ 4 + 9 + 0 - 4 = 9 \]

### Final Answer
Thus, the value of \(a^2 + b^2 + c^2 - 4\) is:

\[ \boxed{9} \]